The ability to propagate the uncertainty information during image processing can be very important in different applications. Detecting edges are an important pre-processing step in image analysis. Best results of image analysis extremely depend on edge detection. Edge detectors are intended to detect and localize the boundaries or silhouettes of objects appearing in images. Up to now many edge detection methods have been developed. But it may have some weaknesses in correct detection of the scope of complications for aerial images or medical images, because of the high variation rate in these images. This paper introduces a verification framework to detect edges based on interval techniques using measuring diversity of pixel's intensity and randomness of intensity distribution within the framework of information theory.
In image processing tasks, there are various sources of ambiguity and uncertainty to be
considered when performing the processing [
Interval arithmetic is a powerful tool to deal with the uncertain data, the concepts of interval
arithmetic are discussed in [
The concept of interval analysis is to compute with intervals of real numbers instead of real
numbers and it considers a powerful tool to determine the effects of uncertain data [
IV(L) = [ (0,
I(x,y) −1)] IV(U) = [
(255, I(x,y) + 1)] IV(M) = [
IV(L)+IV(U)] 2 (1) (2) (3) Proceedings of the 27th International Symposium Nuclear Electronics and Computing (NEC’2019)
That is, we assign to each image position an interval as IV(L) and IV(U) which include all of the brightness values modified by ± 1 tone and IV(M) is the midpoint image of an interval images IV(L) and IV(U) where the brightness values can be modified by α interval operator as 0 < α < 1. So, once we have interval representation images, then we can apply different strategies of computing, as, we can apply the computing strategies individually for IV(L), IV(U) and IV(M) images or together. Figure 1, includes an example of an image, together with the verification interval-valued representation. (4) (5) (6) (7) steps can be observed. Original image (a) is divided into two parts (Upper bound IV(U) image (b) and lower bound IV(L) image (c)) following the midpoint IV(M) image (d)
The concept of entropy become increasingly important in image processing, when an image
often obtained from the probability distribution. The Shannon entropy [
k ∑i=1 pi = 1, where k is the total number of states. The entropy of a discrete source is Shannon entropy has the extensive property (additively) S(X + Y) = S(X) + S(Y).
( ) = − ∑
=1 ( ) Sα =
1 1−α (1 − ∑i=1 piα), z where the real number α is an entropic index that characterizes the degree of non-extensivity. This expression recovers to Shannon entropy in the limit α →1 .
For an image with k gray-levels, let p1, p2, … . , pt, pt+1, … . , pkbe its probability distribution, where pt is the normalized histogram (i.e.,pt = ht⁄(M × N)) and ht is the gray level histogram. Using this distribution, we can derive two probability distributions, one for the object (class A) and the other for the background (class B), as follows:
Sα(X + Y) = Sα(X) +Sα(Y) + (1- α) ∙ Sα(X) ∙ Sα(Y). : 1 , 2 , … . . , , : +1 , +2 , … . . , ,
=
∑ =1 , = ∑ = +1 (6) where is the threshold value.
SX(t) = − ∑ pX ln pX , SY(t) = − ∑ pY ln pY i=1
t
t 1 α − 1
z i=t+1
1 α − 1
z ( 1 − ∑ pYα) (7) (8) SαX(t) = ( 1 − ∑ pXα) , SαY(t) =
i=1 i=t+1 Tsallis entropy SαX(t) is parametrically dependent upon the threshold value for the foreground and background. When Sα(t) is maximized, the luminance level that maximizes the function is considered to be the optimum threshold value.
∗ = Arg max [SαX(t) + SαY(t) + (1 − α) ∙ SαX(t) ∙ SαY(t)]. (9) For edge detection, a spatial filter mask is defined as a matrix w of size × .The process of spatial filtering consists simply of moving a filter mask w of order × from point to point in an image. Assuming that = 2 + 1, = 2 + 1, the smallest meaningful size of the mask is 3 × 3. By moving the window through the whole binary image, the probability of each central pixel of the window can be determined by entropy. Thus, if the gray level of all pixels under the window are homogeneous, = 1, = 0. In this situation, the central pixel is not an edge pixel. In cases, where ≤6/9, the variety of gray level of pixels under the window high, and thus we can assume that we are on an edge pixel.
In order to assess and evaluate the performance of the proposed method, experiments have been performed on the aerial dataset. The performance of the proposed method is assessed qualitatively.
(a)
(b) Figure 2. Samples of aerial images (c)
The data set of aerial images are shown in Figure 2. The subjective comparison of results for the proposed technique of different version of IV images are shown in Figure 3. The results indicate that the proposed technique give a good performance in detecting the edges through consideration of interval concept, which an edges detected have been improved in terms of visual comparison and the boundaries of the objects are more clear in the results of IV images. The results of proposed technique proves that considering the interval arithmetic in designing solutions for some applications may impact the performance of algorithms.
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